\newcommand{\VEC}{\operatorname{vec}}
\newcommand{\vecc}{\operatorname{vec}}
\newcommand{\svec}{\operatorname{svec}}
\newcommand{\cond}{\operatorname{cond}}
\newcommand{\cone}{\operatorname{cone}}
\newcommand{\conv}{\operatorname{conv}}
\newcommand{\argmin}{\operatorname{argmin}}
\newcommand{\rk}{\operatornamewithlimits{rk}}
\newcommand{\range}{\operatorname{Range}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Null}{\operatorname{Null}}
\newcommand{\codim}{\operatorname{co-dim}}
\newcommand{\Mod}{\operatorname{Mod}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\jrk}[1]{\rk_{#1}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\Arw}{\operatorname{Arw}}
\newcommand{\Ocomm}{C_{\J}}
\newcommand{\Comm}{\mathfrak{C}}
\newcommand{\bigO}{\mathcal{O}}
\newcommand{\Cn}{\mathcal{C}_n}
\newcommand{\defeq}{\stackrel{\mathrm{def}}{=}}
\newcommand{\pd}{\succ}
\newcommand{\xbar}{\bar{x}}
%\newcommand{\Th}[1]{$#1^\mathrm{th}$}
\newcommand{\Th}[1]{#1\textsuperscript{th}}
\newcommand{\ubar}{\bar{u}}
\newcommand{\tlup}[1]{\widetilde{#1}}
\newcommand{\tldn}[1]{\undertilde{#1}}
%\newcommand{\tlup}[1]{\tilde{#1}}
%\newcommand{\tldn}[1]{\undertilde{#1}}
\newcommand{\bd}{\operatorname{bd}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\Int}{\operatorname{Int}}
\newcommand{\sgn}{\operatornamewithlimits{sgn}}
\newcommand{\norminf}[1]{\left\|{#1}\right\|_\infty}
\newcommand{\normone}[1]{\left\|{#1}\right\|_1}
\newcommand{\normp}[1]{\left\|{#1}\right\|_p}
\newcommand{\normq}[1]{\left\|{#1}\right\|_q}
\newcommand{\normtwo}[1]{\left\|{#1}\right\|_2}
\newcommand{\normF}[1]{\left\|{#1}\right\|_{\mathrm{F}}}
\newcommand{\jnormF}[2]{\left\| {#1}\right\|_{\mathrm{F/{#2}}}}
\newcommand{\scp}[2]{\langle{#1},{#2}\rangle}
\newcommand{\jscp}[3]{\langle{#1},{#2}\rangle_{#3}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\trace}{\operatorname{trace}}
\newcommand{\jtr}[1]{\tr_{#1}}
\newcommand{\Det}{\operatorname{Det}}
\newcommand{\jdet}[1]{\det_{#1}}
\newcommand{\Tr}{\operatorname{Trace}}
\newcommand{\Trace}{\operatorname{Trace}}
\newcommand{\op}[1]{{#1}^{\text{old}}} %old solution
\newcommand{\np}[1]{{#1}^{\text{new}}} %new solution
\newcommand{\Qp}[1]{\mathbf{Q}_{#1}}
\newcommand{\jprod}{\circ}\newcommand{\vzero}{\mathbf{0}}

\newcommand{\PR}{&\propto&}

\newcommand{\vone}{\mathbf{1}}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\vd}{\mathbf{d}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vf}{\mathbf{f}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\vh}{\mathbf{h}}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\newcommand{\vl}{\mathbf{l}}
\newcommand{\vm}{\mathbf{m}}
\newcommand{\vn}{\mathbf{n}}
\newcommand{\vo}{\mathbf{o}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vq}{\mathbf{q}}
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vs}{\mathbf{s}}
\newcommand{\vt}{\mathbf{t}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vz}{\mathbf{z}}
\newcommand{\valpha}{\boldsymbol{\alpha}}
\newcommand{\vbeta}{\boldsymbol{\beta}}
\newcommand{\vgamma}{\boldsymbol{\gamma}}
\newcommand{\vdelta}{\boldsymbol{\delta}}
\newcommand{\vepsilon}{\boldsymbol{\epsilon}}
\newcommand{\vvarepsilon}{\boldsymbol{\varepsilon}}
\newcommand{\vzeta}{\boldsymbol{\zeta}}
\newcommand{\veta}{\boldsymbol{\eta}}
\newcommand{\vtheta}{\boldsymbol{\theta}}
\newcommand{\vTheta}{\boldsymbol{\Theta}}
\newcommand{\vPsi}{\boldsymbol{\Psi}}
\newcommand{\vvartheta}{\boldsymbol{\vartheta}}
\newcommand{\viota}{\boldsymbol{\iota}}
\newcommand{\vkappa}{\boldsymbol{\kappa}}
\newcommand{\vlambda}{\boldsymbol{\lambda}}
\newcommand{\vLambda}{\boldsymbol{\lambda}}
\newcommand{\vmu}{\boldsymbol{\mu}}
\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vxi}{\boldsymbol{\xi}}
\newcommand{\vomicron}{\boldsymbol{o}}
\newcommand{\vpi}{\boldsymbol{\pi}}
\newcommand{\vvarpi}{\boldsymbol{\varpi}}
\newcommand{\vrho}{\boldsymbol{\rho}}
\newcommand{\vvarrho}{\boldsymbol{\varrho}}
\newcommand{\vsigma}{\boldsymbol{\sigma}}
\newcommand{\vvarsigma}{\boldsymbol{\varsigma}}
\newcommand{\vtau}{\boldsymbol{\tau}}
\newcommand{\vupsilon}{\boldsymbol{\upsilon}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\vvarphi}{\boldsymbol{\varphi}}
\newcommand{\vchi}{\boldsymbol{\chi}}
\newcommand{\vpsi}{\boldsymbol{\psi}}
\newcommand{\vomega}{\boldsymbol{\omega}}

\newcommand{\bva}{\overline{\mathbf{a}}}
\newcommand{\bvb}{\overline{\mathbf{b}}}
\newcommand{\bvc}{\overline{\mathbf{c}}}
\newcommand{\bvd}{\overline{\mathbf{d}}}
\newcommand{\bve}{\overline{\mathbf{e}}}
\newcommand{\bvf}{\overline{\mathbf{f}}}
\newcommand{\bvg}{\overline{\mathbf{g}}}
\newcommand{\bvh}{\overline{\mathbf{h}}}
\newcommand{\bvi}{\overline{\mathbf{i}}}
\newcommand{\bvj}{\overline{\mathbf{j}}}
\newcommand{\bvk}{\overline{\mathbf{k}}}
\newcommand{\bvl}{\overline{\mathbf{l}}}
\newcommand{\bvm}{\overline{\mathbf{m}}}
\newcommand{\bvn}{\overline{\mathbf{n}}}
\newcommand{\bvo}{\overline{\mathbf{o}}}
\newcommand{\bvp}{\overline{\mathbf{p}}}
\newcommand{\bvq}{\overline{\mathbf{q}}}
\newcommand{\bvr}{\overline{\mathbf{r}}}
\newcommand{\bvs}{\overline{\mathbf{s}}}
\newcommand{\bvt}{\overline{\mathbf{t}}}
\newcommand{\bvu}{\overline{\mathbf{u}}}
\newcommand{\bvv}{\overline{\mathbf{v}}}
\newcommand{\bvw}{\overline{\mathbf{w}}}
\newcommand{\bvx}{\overline{\mathbf{x}}}
\newcommand{\bvy}{\overline{\mathbf{y}}}
\newcommand{\bvz}{\overline{\mathbf{z}}}

\newcommand{\bvA}{{\mathbf{A}}}
\newcommand{\bvB}{{\mathbf{B}}}
\newcommand{\bvC}{{\mathbf{C}}}
\newcommand{\bvD}{{\mathbf{D}}}
\newcommand{\bvE}{{\mathbf{E}}}
\newcommand{\bvF}{{\mathbf{F}}}
\newcommand{\bvG}{{\mathbf{G}}}
\newcommand{\bvH}{{\mathbf{H}}}
\newcommand{\bvI}{{\mathbf{I}}}
\newcommand{\bvJ}{{\mathbf{J}}}
\newcommand{\bvK}{{\mathbf{K}}}
\newcommand{\bvL}{{\mathbf{L}}}
\newcommand{\bvM}{{\mathbf{M}}}
\newcommand{\bvN}{{\mathbf{N}}}
\newcommand{\bvO}{{\mathbf{O}}}
\newcommand{\bvP}{{\mathbf{P}}}
\newcommand{\bvQ}{{\mathbf{Q}}}
\newcommand{\bvR}{{\mathbf{R}}}
\newcommand{\bvS}{{\mathbf{S}}}
\newcommand{\bvT}{{\mathbf{T}}}
\newcommand{\bvU}{{\mathbf{U}}}
\newcommand{\bvV}{{\mathbf{V}}}
\newcommand{\bvW}{{\mathbf{W}}}
\newcommand{\bvX}{{\mathbf{X}}}
\newcommand{\bvY}{{\mathbf{Y}}}
\newcommand{\bvZ}{{\mathbf{Z}}}

\newcommand{\hva}{{\mathbf{\hat a}}}
\newcommand{\hvb}{{\mathbf{\hat b}}}
\newcommand{\hvc}{{\mathbf{\hat c}}}
\newcommand{\hvd}{{\mathbf{\hat d}}}
\newcommand{\hve}{{\mathbf{\hat e}}}
\newcommand{\hvf}{{\mathbf{\hat f}}}
\newcommand{\hvg}{{\mathbf{\hat g}}}
\newcommand{\hvh}{{\mathbf{\hat h}}}
\newcommand{\hvi}{{\mathbf{\hat i}}}
\newcommand{\hvj}{{\mathbf{\hat j}}}
\newcommand{\hvk}{{\mathbf{\hat k}}}
\newcommand{\hvl}{{\mathbf{\hat l}}}
\newcommand{\hvm}{{\mathbf{\hat m}}}
\newcommand{\hvn}{{\mathbf{\hat n}}}
\newcommand{\hvo}{{\mathbf{\hat o}}}
\newcommand{\hvp}{{\mathbf{\hat p}}}
\newcommand{\hvq}{{\mathbf{\hat q}}}
\newcommand{\hvr}{{\mathbf{\hat r}}}
\newcommand{\hvs}{{\mathbf{\hat s}}}
\newcommand{\hvt}{{\mathbf{\hat t}}}
\newcommand{\hvu}{{\mathbf{\hat u}}}
\newcommand{\hvv}{{\mathbf{\hat v}}}
\newcommand{\hvw}{{\mathbf{\hat w}}}
\newcommand{\hvx}{{\mathbf{\hat x}}}
\newcommand{\hvy}{{\mathbf{\hat y}}}
\newcommand{\hvz}{{\mathbf{\hat z}}}

\newcommand{\xt}{\tlup{\vx}}
\newcommand{\wt}{\tlup{\vw}}
\newcommand{\st}{\tldn{\vs}}
\newcommand{\ct}{\tldn{\vc}}
\newcommand{\at}{\tldn{\va_i}}
\newcommand{\At}{\tldn{A}}
\newcommand{\Xt}{\tilde{X}}
\newcommand{\St}{\tldn{S}}
\newcommand{\dXt}{\widetilde{\dX}}
\newcommand{\dSt}{\tldn{\dS}}
\newcommand{\dxt}{\tlup{\dx}}
\newcommand{\dst}{\tldn{\ds}}
\newcommand{\onehalf}{{1\!/\!2}}
\newcommand{\nonehalf}{{-\!1\!/\!2}}
\newcommand{\xtsq}{\xt^{\onehalf}}
\newcommand{\xtisq}{\xt^{\nonehalf}}
\newcommand{\stsq}{\st^{\onehalf}}
\newcommand{\stisq}{\st^{\nonehalf}}
\newcommand{\Xtsq}{\Xt^{\onehalf}}
\newcommand{\Xtisq}{\Xt^{\nonehalf}}
\newcommand{\Stsq}{\St^{\onehalf}}
\newcommand{\Stisq}{\tldn{S}^{\nonehalf}}
\newcommand{\Xsq}{X^{\onehalf}}
\newcommand{\Xisq}{X^{\nonehalf}}
\newcommand{\Ssq}{S^{\onehalf}}
\newcommand{\Sisq}{S^{\nonehalf}}
\newcommand{\xsq}{\vx^{\onehalf}}
\newcommand{\xisq}{\vx^{\nonehalf}}
\newcommand{\ssq}{\vs^{\onehalf}}
\newcommand{\sisq}{\vs^{\nonehalf}}

\newcommand{\dx}{\Delta \vx}
\newcommand{\dy}{\Delta \vy}
\newcommand{\ds}{\Delta \vs}
\newcommand{\sh}{\hat{\vs}}



\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\Bn}{\mathcal{B}_n}
\newcommand{\C}{\mathcal{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\G}{\mathcal{G}}
\newcommand{\Gp}{\mathcal{G}_+}
\renewcommand{\H}{\mathcal{H}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\Lo}{\mathcal{L}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\Po}{\mathcal{P}}
\newcommand{\Q}{\mathcal{Q}}
\newcommand{\Oc}{\mathcal{O}}
\newcommand{\Sy}{\mathcal{S}}
\newcommand{\T}{\mathcal{T}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\W}{\mathcal{W}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\Z}{\mathcal{Z}}

\newcommand{\matA}{\mathfrak{A}}
\newcommand{\matB}{\mathfrak{B}}
\newcommand{\matC}{\mathfrak{C}}
\newcommand{\matD}{\mathfrak{D}}
\newcommand{\matE}{\mathfrak{E}}
\newcommand{\matF}{\mathfrak{F}}
\newcommand{\matG}{\mathfrak{G}}
\newcommand{\matH}{\mathfrak{H}}
\newcommand{\matI}{\mathfrak{I}}
\newcommand{\matJ}{\mathfrak{J}}
\newcommand{\matK}{\mathfrak{K}}
\newcommand{\matL}{\mathfrak{L}}
\newcommand{\matM}{\mathfrak{M}}
\newcommand{\matN}{\mathfrak{N}}
\newcommand{\matO}{\mathfrak{O}}
\newcommand{\matP}{\mathfrak{P}}
\newcommand{\matQ}{\mathfrak{Q}}
\newcommand{\matR}{\mathfrak{R}}
\newcommand{\matS}{\mathfrak{S}}
\newcommand{\matT}{\mathfrak{T}}
\newcommand{\matU}{\mathfrak{U}}
\newcommand{\matV}{\mathfrak{V}}
\newcommand{\matW}{\mathfrak{W}}
\newcommand{\matX}{\mathfrak{X}}
\newcommand{\matY}{\mathfrak{Y}}
\newcommand{\matZ}{\mathfrak{Z}}

\newcommand{\bbA}{\mathbb{A}}
\newcommand{\bbB}{\mathbb{B}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbD}{\mathbb{D}}
\newcommand{\bbE}{\mathbb{E}}
\newcommand{\bbF}{\mathbb{F}}
\newcommand{\bbG}{\mathbb{G}}
\newcommand{\bbH}{\mathbb{H}}
\newcommand{\bbI}{\mathbb{I}}
\newcommand{\bbJ}{\mathbb{J}}
\newcommand{\bbK}{\mathbb{K}}
\newcommand{\bbL}{\mathbb{L}}
\newcommand{\bbM}{\mathbb{M}}
\newcommand{\bbN}{\mathbb{N}}
\newcommand{\bbO}{\mathbb{O}}
\newcommand{\bbP}{\mathbb{P}}
\newcommand{\bbQ}{\mathbb{Q}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbS}{\mathbb{S}}
\newcommand{\bbT}{\mathbb{T}}
\newcommand{\bbU}{\mathbb{U}}
\newcommand{\bbV}{\mathbb{V}}
\newcommand{\bbW}{\mathbb{W}}
\newcommand{\bbX}{\mathbb{X}}
\newcommand{\bbY}{\mathbb{Y}}
\newcommand{\bbZ}{\mathbb{Z}}

\newcommand{\real}{\mathbb{R}}
\newcommand{\realS}{\mathbb{S}}
\newcommand{\realH}{\mathbb{H}}
\newcommand{\realQ}{\mathbb{Q}}
\newcommand{\realO}{\mathbb{O}}

\newcommand{\ST}{{\text{s.t.}}}
\newcommand{\NF}[1]{\N_{F/{#1}}}
\newcommand{\Quart}{\mbox{\boldmath H}}
\newcommand{\psd}{\succcurlyeq}
\newcommand{\pdef}{\succ}
%\newcommand{\pd}{\succ} %use for strict semidefinite inequality
\newcommand{\Qgeq}{\succcurlyeq_\Q}
\newcommand{\Qgt}{\succ_\Q}
\newcommand{\Kgeq}{\succcurlyeq_\K}
\newcommand{\Kgt}{\succ_\K}
\newcommand{\skron}{\circledast}
\newcommand{\kron}{\otimes}
\newcommand{\Diag}{\operatorname{Diag}}
\newcommand{\diag}{\mathbf{diag}} %diag
\newcommand{\dt}{\Delta} %Delta
\DeclareMathOperator*{\Bd}{bd} %For boundary
\DeclareMathOperator*{\Cl}{cl}%For closure
%Use the following for definitions lemmas etc.

\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{note}{Note}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{claim}{Claim}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}
\newtheorem{corollary}{Corollary}
\newtheorem{observation}{Observation}
\newtheorem{fact}{Fact}
\newenvironment{Example}[1]{\begin{example}[#1]\rm}{{\hfill\rule{2mm}{2mm}\par}\end{example}}
\newenvironment{Remark}{\begin{remark}\rm}{{\hfill\rule{2mm}{2mm}\par}\end{remark}}
\newenvironment{proof}{{\noindent \textbf{Proof:}}}{\hfill\rule{2mm}{2mm}\par}
%
